Prove by induction that, for all $n\in\Bbb N$, $\sqrt{n} ≤ \sum_ {k=1}^n \frac{1}{\sqrt{k}} < \sqrt{n} + \frac{n}{\sqrt{n+1}}$.

Question

So, I know that for my base case I use $n=1$, and that for the inductive hypothesis we assume the pattern holds until the $n-th$ iteration. Then use that to prove the $(n+1)-th$ iteration ($\Bbb P(n)\implies\Bbb P(n+1)$). The part that is giving me the most trouble is how to actually use the inductive hypothesis given that you're dealing with two inequalities. The algebra of having so many square-roots is also giving me a hard time if you know any hints. Thanks.

Answer 1

If you know $\sqrt{n} \le \sum_{k=1}^n \frac{1}{\sqrt{k}}$ then adding $\frac{1}{\sqrt{n+1}}$ to both sides yields $$\sqrt{n} + \frac{1}{\sqrt{n+1}} \le \sum_{k=1}^{n+1} \frac{1}{\sqrt{k}}.$$ It then suffices to show $$\sqrt{n+1} \le \sqrt{n} + \frac{1}{\sqrt{n+1}},$$ which I leave to you.

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If you know $\sum_{k=1}^n \frac{1}{\sqrt{k}} < \sqrt{n} + \frac{n}{\sqrt{n+1}}$ then adding $\frac{1}{\sqrt{n+1}}$ to both sides yields $$\sum_{k=1}^{n+1} \frac{1}{\sqrt{k}} < \sqrt{n} + \frac{n + 1}{\sqrt{n+1}} = \sqrt{n} + \sqrt{n+1}.$$ It then suffices to show $$\sqrt{n} \le \frac{n+1}{\sqrt{n+2}},$$ which I leave to you.